Convex bodies: the brunn–minkowski theory

Chapter 15 – Geometric Shortest Paths and Network Optimization

News articles contain a wealth of implicit geographic content that if exposed to readers improves understanding of today's news. However, most articles are not explicitly geotagged with their geographic content, and few news aggregation systems expose this content to users. A new system named NewsStand is presented that collects, analyzes, and displays news stories in a map interface, thus leveraging on their implicit geographic content. NewsStand monitors RSS feeds from thousands of online news sources and retrieves articles within minutes of publication. It then extracts geographic content from articles using a custom-built geotagger, and groups articles into story clusters using a fast online clustering algorithm. By panning and zooming in NewsStand's map interface, users can retrieve stories based on both topical significance and geographic region, and see substantially different stories depending on position and zoom level.

NewsStand: a new view on news

We address the problem of filtering, selecting and placing labels on a dynamic map, which is characterized by continuous zooming and panning capabilities. This consists of two interrelated issues. The first is to avoid label popping and other artifacts that cause confusion and interrupt navigation, and the second is to label at interactive speed. In most formulations the static map labeling problem is NP-hard, and a fast approximation might have O(n log n) complexity. Even this is too slow during interaction, when the number of labels shown can be several orders of magnitude less than the number in the map. In this paper we introduce a set of desiderata for "consistent" dynamic map labeling, which has qualities desirable for navigation. We develop a new framework for dynamic labeling that achieves the desiderata and allows for fast interactive display by moving all of the selection and placement decisions into the preprocessing phase. This framework is general enough to accommodate a variety of selection and placement algorithms. It does not appear possible to achieve our desiderata using previous frameworks. Prior to this paper, there were no formal models of dynamic maps or of dynamic labels; our paper introduces both. We formulate a general optimization problem for dynamic map labeling and give a solution to a simple version of the problem. The simple version is based on label priorities and a versatile and intuitive class of dynamic label placements we call "invariant point placements". Despite these restrictions, our approach gives a useful and practical solution. Our implementation is incorporated into the G-Vis system which is a full-detail dynamic map of the continental USA. This demo is available through any browser

Dynamic Map Labeling

The survey contains results related to different aspects of polyhedral approximation of convex bodies and some adjacent problems.

Approximation of convex sets by polytopes

We investigate several straight-line drawing problems for bounded-degree trees in the integer grid without edge crossings under various types of drawings: (1) upward drawings whose edges are drawn as vertically monotone chains, a sequence of line segments, from a parent to its children, (2) order-preserving drawings which preserve the left-to-right order of the children of each vertex, and (3) orthogonal straight-line drawings in which each edge is represented as a single vertical or horizontal segment. Main contribution of this paper is a unified framework to reduce the upper bound on area for the straight-line drawing problems from O (n log n) (Crescenzi et al., 1992) to O (n loglog n) . This is the first solution of an open problem stated by Garg et al. (1993). We also show that any binary tree admits a small area drawing satisfying any given aspect ratio in the orthogonal straight-line drawing type. Our results are briefly summarized as follows. Let T be a bounded-degree tree with n vertices. Firstly, we show that T admits an upward straight-line drawing with area O (n loglog n) . If T is binary, we can obtain an O (n loglog n) -area upward orthogonal drawing in which each edge is drawn as a chain of at most two orthogonal segments and which has O (n/ log n) bends in total. Secondly, we present O (n loglog n) -area (respectively, -volume) orthogonal straight-line drawing algorithms for binary trees with arbitrary aspect ratios in 2-dimension (respectively, 3-dimension). Finally, we present some experimental results which shows the area requirements, in practice, for (order-preserving) upward drawing are much smaller than theoretical bounds obtained through analysis.

Area-efficient algorithms for straight-line tree drawings

A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a well-known graph-drawing problem.

Further we define fix(G,δ)=n−shift(G,δ) to be the maximum number of vertices of a planar n-vertex graph G that can be fixed when untangling δ. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log\log n}$vertices when untangling a drawing of an n-vertex graph G. If G is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$vertices. On the other hand, we construct, for arbitrarily large n, an n-vertex planar graph G and a drawing δ G of G with $\ensuremath {\mathrm {fix}}(G,\delta_{G})\leq \sqrt{n-2}+1$and an n-vertex outerplanar graph H and a drawing δ H of H with $\ensuremath {\mathrm {fix}}(H,\delta_{H})\leq2\sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

/pdf/untangling-a-planar-graph-1q3miptpu7.pdf

Untangling a Planar Graph

Annotating maps, graphs, and diagrams with pieces of text is an
 important step in information visualization that is usually referred
 to as label placement. We define nine label-placement models for
 labeling points with axis-parallel rectangles given a weight for
 each point. There are two groups: fixed-position models and slider
 models. We aim to maximize the weight sum of those points that
 receive a label.

 We first compare our models by giving bounds for the ratios between
 the weights of maximum-weight labelings in different models. Then
 we present algorithms for labeling n points with unit-height
 rectangles. We show how an O(n\log n)-time factor-2 approximation
 algorithm and a PTAS for fixed-position models can be extended to
 handle the weighted case. Our main contribution is the first
 algorithm for weighted sliding labels. Its approximation factor is
 (2+\varepsilon), it runs in O(n
 2/\varepsilon) time and uses
 O(n/\varepsilon) space. We show that other than for fixed-position models even the
 projection to one dimension remains NP-hard.

 For slider models we also investigate some special cases, namely
 (a) the number of different point weights is bounded, (b) all labels
 are unit squares, and (c) the ratio between maximum and minimum
 label height is bounded.

Labeling Points with Weights

Given two convex polyhedra P and Q in three-dimensional space, we consider two related problems of shape matching: (1) finding a translation t"1@?R^3 of Q that maximizes the volume of their overlap P@?(Q+t"1), and (2) finding a translation t"2@?R^3 that minimizes the volume of the convex hull of P@?(Q+t"2). For the maximum overlap problem, we observe that the dth root of the objective function is concave and present an algorithm that computes the optimal translation in expected time O(n^3log^4n). This method generalizes to higher dimensions d>3 with expected running time O(n^d^+^1^-^3^d(logn)^d^+^1). For the minimum convex hull problem, we show that the objective function is convex. The same method used for the maximum overlap problem can be applied to this problem and the optimal translation can be computed in the same time bound.

Maximum overlap and minimum convex hull of two convex polyhedra under translations

Given two compact convex sets P and Q in the plane, we compute an image of P under a rigid motion that approximately maximizes the overlap with Q. More precisely, for any @e>0, we compute a rigid motion such that the area of overlap is at least [email protected] times the maximum possible overlap. Our algorithm uses O(1/@e) extreme point and line intersection queries on P and Q, plus O((1/@e^2)log(1/@e)) running time. If only translations are allowed, the extra running time reduces to O((1/@e)log(1/@e)). If P and Q are convex polygons with n vertices in total that are given in an array or balanced tree, the total running time is O((1/@e)logn+(1/@e^2)log(1/@e)) for rigid motions and O((1/@e)logn+(1/@e)log(1/@e)) for translations.

Chan-Su Shin

Papers

Area-efficient algorithms for straight-line tree drawings

Untangling a Planar Graph

Labeling Points with Weights

Maximum overlap and minimum convex hull of two convex polyhedra under translations

Maximizing the overlap of two planar convex sets under rigid motions